{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Stats\n",
    "\n",
    "This notebook loads some data, reports some simple descriptive statistics (means, standard deviations etc) and shows a number of useful plots (scatter plots, histograms, time series plots).\n",
    "\n",
    "Most of the descriptive stats use the standard package Statistics. The plots rely on the Plots package and the pdf and quantiles are from the [Distributions](https://github.com/JuliaStats/Distributions.jl) package.\n",
    "\n",
    "For more stat functions, see the [StatsBase](https://github.com/JuliaStats/StatsBase.jl) package. (Not used here.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load Packages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "printTable2"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Statistics, Dates, LinearAlgebra, DelimitedFiles, Distributions\n",
    "\n",
    "include(\"printmat.jl\")     \n",
    "include(\"printTable.jl\") #a function for prettier table printing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "\n",
    "#pyplot(size=(600,400))\n",
    "gr(size=(480,320))\n",
    "default(fmt = :svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Load Data from a csv File\n",
    "\n",
    "The following is a portion of MyData.csv:\n",
    "\n",
    "```\n",
    "date,Mkt-RF,RF,SmallGrowth\n",
    "197901,4.18,0.77,10.96\n",
    "197902,-3.41,0.73,-2.09\n",
    "197903,5.75,0.81,11.71\n",
    "197904,0.05,0.8,3.27\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "first four lines of x:\n",
      "197901.000     4.180     0.770    10.960\n",
      "197902.000    -3.410     0.730    -2.090\n",
      "197903.000     5.750     0.810    11.710\n",
      "197904.000     0.050     0.800     3.270\n",
      "\n"
     ]
    }
   ],
   "source": [
    "x = readdlm(\"Data/MyData.csv\",',',skipstart=1)  #reading the csv file\n",
    "                                                #skip 1st line\n",
    "println(\"\\nfirst four lines of x:\")\n",
    "printmat(x[1:4,:])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Creating Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "first 4 obs of Rme and Re\n",
      "     4.180    10.190\n",
      "    -3.410    -2.820\n",
      "     5.750    10.900\n",
      "     0.050     2.470\n",
      "\n",
      "1979-01-01\n",
      "1979-02-01\n",
      "1979-03-01\n",
      "1979-04-01\n",
      "\n"
     ]
    }
   ],
   "source": [
    "ym  = round.(Int,x[:,1])             #yearmonth, like 200712\n",
    "(Rme,Rf,R) = (x[:,2],x[:,3],x[:,4])  #creating variables from columns of x\n",
    "Re  = R - Rf                         #do R .- Rf if R has several columns\n",
    "\n",
    "println(\"first 4 obs of Rme and Re\")\n",
    "printmat([Rme[1:4,:] Re[1:4,:]])\n",
    "\n",
    "dN = Date.(string.(ym),\"yyyymm\") #convert to string and then to Julia Date\n",
    "printmat(dN[1:4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Some Descriptive Statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Means and Standard Deviations\n",
    "\n",
    "The next few cells estimate means, standard deviations, covariances and correlations of the variables `Rme` (US equity market excess return) and `Re` (excess returns for a segment of the market, small growth firms). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "           Rme        Re\n",
      "mean     0.602     0.303\n",
      "std      4.604     8.572\n",
      "\n"
     ]
    }
   ],
   "source": [
    "μ = mean([Rme Re],dims=1)    #,dims=1 to calculate average along a column\n",
    "σ = std([Rme Re],dims=1)     #do \\sigma[Tab] to get σ\n",
    "\n",
    "printTable([μ;σ],[\"Rme\",\"Re\"],[\"mean\",\"std\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Covariances and Correlations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "cov([Rme Re]): \n",
      "    21.197    28.426\n",
      "    28.426    73.475\n",
      "\n",
      "\n",
      "cor([Rme Re]): \n",
      "     1.000     0.720\n",
      "     0.720     1.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "println(\"\\n\",\"cov([Rme Re]): \")\n",
    "printmat(cov([Rme Re]))\n",
    "\n",
    "println(\"\\n\",\"cor([Rme Re]): \")\n",
    "printmat(cor([Rme Re]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## OLS\n",
    "\n",
    "A linear regression\n",
    "$\n",
    "y_t = x_t'b + u_t\n",
    "$,\n",
    "where $x_t=[1;R^e_{m,t}]$.\n",
    "\n",
    "Clearly, the first element of $b$ is the intercept and the second element is the slope coefficient.\n",
    "\n",
    "The code below creates a $Tx2$ matrix `x` with $x'_t$ on row $t$. $y$ is a T-element vector (or $Tx1$).\n",
    "\n",
    "The GLM package (not used here) has powerful regression methods. See https://github.com/JuliaStats/GLM.jl."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "OLS coefficients, regressing Re on constant and Rme\n",
      "        Calc1     Calc2\n",
      "c      -0.504    -0.504\n",
      "Rme     1.341     1.341\n",
      "\n",
      "      R2:      0.519\n",
      "no. of observations:        388\n"
     ]
    }
   ],
   "source": [
    "c   = ones(size(Rme,1))         #a vector with ones, no. rows from variable\n",
    "x   = [c Rme]                   #x is a Tx2 matrix\n",
    "y   = copy(Re)                  #to get standard OLS notation\n",
    "\n",
    "b2  = inv(x'x)*x'y              #OLS according to a textbook\n",
    "b   = x\\y                       #also OLS, quicker and numerically more stable\n",
    "u   = y - x*b                   #OLS residuals\n",
    "R2a = 1 - var(u)/var(y)         #R2, but that name is already taken\n",
    "\n",
    "println(\"OLS coefficients, regressing Re on constant and Rme\")\n",
    "printTable([b b2],[\"Calc1\",\"Calc2\"],[\"c\",\"Rme\"])\n",
    "printlnPs(\"R2: \",R2a) \n",
    "printlnPs(\"no. of observations: \",size(Re,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "coeffs and t-stats\n",
      "         Coef    t-stat\n",
      "c      -0.504    -1.656\n",
      "Rme     1.341    20.427\n",
      "\n"
     ]
    }
   ],
   "source": [
    "Covb  = inv(x'x)*var(u)         #covariance matrix of b estimates\n",
    "Stdb  = sqrt.(diag(Covb))       #std of b estimates\n",
    "tstat = (b .- 0)./Stdb          #t-stats, replace 0 with your null hypothesis \n",
    "\n",
    "println(\"\\ncoeffs and t-stats\")\n",
    "printTable([b tstat],[\"Coef\",\"t-stat\"],[\"c\",\"Rme\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Drawing Random Numbers and Finding Critical Values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random Numbers: Independent Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "mean and std of random draws: \n",
      "      series 1  series 2\n",
      "mean    -0.082     0.139\n",
      "std      1.078     0.930\n",
      "\n",
      "covariance matrix:\n",
      "     1.161     0.078\n",
      "     0.078     0.865\n",
      "\n",
      "correlation matrix:\n",
      "     1.000     0.078\n",
      "     0.078     1.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "T = 100\n",
    "x = randn(T,2)    #T x 2 matrix, N(0,1) distribution\n",
    "\n",
    "println(\"\\n\",\"mean and std of random draws: \")\n",
    "μ = mean(x,dims=1)\n",
    "σ = std(x,dims=1)\n",
    "printTable([μ;σ],[\"series 1\",\"series 2\"],[\"mean\",\"std\"])\n",
    "\n",
    "println(\"covariance matrix:\")\n",
    "printmat(cov(x))\n",
    "println(\"correlation matrix:\")\n",
    "printmat(cor(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Random Numbers: Correlated Variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "covariance matrix\n",
      "     0.866     0.350\n",
      "     0.350     1.817\n",
      "\n",
      "correlation matrix:\n",
      "     1.000     0.279\n",
      "     0.279     1.000\n",
      "\n"
     ]
    }
   ],
   "source": [
    "μ = [-1,10]          #vector of means\n",
    "Σ = [1 0.5;          #covariance matrix\n",
    "     0.5 2]\n",
    "\n",
    "T = 100\n",
    "x = rand(MvNormal(μ,Σ),T)'  #random numbers, T x 2, drawn from bivariate N(μ,Σ)\n",
    "\n",
    "println(\"covariance matrix\")\n",
    "printmat(cov(x))\n",
    "println(\"correlation matrix:\")\n",
    "printmat(cor(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quantiles (\"critical values\") of Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "5th percentile of N(0,1) and 95th of Chisquare(5)\n",
      "    -1.645    11.070\n",
      "\n"
     ]
    }
   ],
   "source": [
    "N05     = quantile(Normal(0,1),0.05)            #from the Distributions package\n",
    "Chisq05 = quantile(Chisq(5),0.95)\n",
    "\n",
    "println(\"\\n\",\"5th percentile of N(0,1) and 95th of Chisquare(5)\")      #lots of statistics functions\n",
    "printmat([N05 Chisq05])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Statistical Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "xTicksLoc = Dates.value.([Date(1980);Date(1990);Date(2000);Date(2010)])\n",
    "xTicksLab = [\"1980\";\"1990\";\"2000\";\"2010\"]   #crude way of getting the tick marks right\n",
    "\n",
    "plot( dN,Rme,\n",
    "      linecolor = :blue,\n",
    "      legend = false,\n",
    "      xticks = (xTicksLoc,xTicksLab),\n",
    "      ylim = (-25,25),\n",
    "      title = \"Time series plot: monthly US equity market excess return\",\n",
    "      titlefont = font(10),\n",
    "      ylabel = \"%\" )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
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     "execution_count": 13,
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   "source": [
    "scatter( Rme,Re,\n",
    "         fillcolor = :blue,\n",
    "         legend = false,\n",
    "         xlim = (-40,40),\n",
    "         ylim = (-40,60),\n",
    "         title = \"Scatter plot: two monthly return series (and 45 degree line)\",\n",
    "         titlefont = font(10),\n",
    "         xlabel = \"Market excess return, %\",\n",
    "         ylabel = \"Excess returns on small growth stocks, %\",\n",
    "         guidefont = font(8) )\n",
    "\n",
    "plot!([-40;60],[-40;60],color=:black,linewidth=0.5)   #easier to keep this outside plot()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
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       "  1074.85,615.458 1078.57,605.101 1082.28,594.753 1086,584.426 1089.71,574.129 1093.43,563.873 1097.15,553.668 1100.86,543.526 1104.58,533.457 1108.29,523.471 \n",
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      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xGrid = -25:0.1:15\n",
    "pdfX  = pdf.(Normal(mean(Rme),std(Rme)),xGrid) #\"Distributions\" wants σ, not σ^2\n",
    "\n",
    "histogram( Rme,bins = -25:1:15,\n",
    "           normalized = true,     #normalized to have area=1\n",
    "           label = \"histogram\",\n",
    "           title = \"Histogram: monthly US equity market excess return\",\n",
    "           titlefont = font(10),\n",
    "           xlabel = \"Market excess return, %\",\n",
    "           legend = :topleft,\n",
    "           guidefont = font(8) )   #font size of axis labels\n",
    "\n",
    "plot!(xGrid,pdfX,linewidth=3,label=\"fitted N()\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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